Using the provided Venn diagram and definitions, identify equivalent sets and calculate the square root of the given numbers using prime factorization.

• Definition of a proper subset: A set $A$ is a proper subset of a set $B$ if all elements of $A$ are also elements of $B$, and $B$ contains at least one element not in $A$. This is denoted as $A \subset B$.
• Elements of sets A, B, and C from the diagram:
  • $A = \{b, c, d, e, f\}$
  • $B = \{d, e, f, g\}$
  • $C = \{f, g, h, i\}$
• Identify equivalent sets: Two sets are equivalent if they have the same number of elements (cardinality).
  • Number of elements in $A$, $|A| = 5$.
  • Number of elements in $B$, $|B| = 4$.
  • Number of elements in $C$, $|C| = 4$.
  • Since $|B| = |C|$, sets $B$ and $C$ are equivalent sets.
  • The equivalent sets are $\boxed{B \text{ and } C}$.

• a) For 50176: $$50176 = 2 \times 25088$$ $$= 2 \times 2 \times 12544$$ $$= 2 \times 2 \times 2 \times 6272$$ $$= 2 \times 2 \times 2 \times 2 \times 3136$$ $$= 2 \times 2 \times 2 \times 2 \times 2 \times 1568$$ $$= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 784$$ $$= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 392$$ $$= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 196$$ $$= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 98$$ $$= 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 49$$ $$= 2^{10} \times 7^2$$ To find the square root, we take half of each exponent: $$\sqrt{50176} = \sqrt{2^{10} \times 7^2} = 2^{10/2} \times 7^{2/2} = 2^5 \times 7^1$$ $$= 32 \times 7 = 224$$ The square root of 50176 is $\boxed{224}$.
• b) For 105625: $$105625 = 5 \times 21125$$ $$= 5 \times 5 \times 4225$$ $$= 5 \times 5 \times 5 \times 845$$ $$= 5 \times 5 \times 5 \times 5 \times 169$$ $$= 5^4 \times 13^2$$ To find the square root, we take half of each exponent: $$\sqrt{105625} = \sqrt{5^4 \times 13^2} = 5^{4/2} \times 13^{2/2} = 5^2 \times 13^1$$ $$= 25 \times 13 = 325$$ The square root of 105625 is $\boxed{325}$.

• a) For 22: The cube number of 22 is $22^3$. $$22^3 = 22 \times 22 \times 22$$ $$= 484 \times 22$$ $$= 10648$$ The cube number of 22 is $\boxed{10648}$.
• b) For 44: The cube number of 44 is $44^3$. $$44^3 = 44 \times 44 \times 44$$ $$= 1936 \times 44$$ $$= 85184$$ The cube number of 44 is $\boxed{85184}$.

100 between Ram and Hari in the ratio 2:3.

• Total amount = Rs. 100
• Ratio of shares for Ram and Hari = 2:3
• Sum of the ratio parts = $2 + 3 = 5$
• Ram's share: $$\text{Ram's share} = \frac{2}{5} \times 100 = 2 \times 20 = 40$$
• Hari's share: $$\text{Hari's share} = \frac{3}{5} \times 100 = 3 \times 20 = 60$$
• Ram gets $\boxed{\text{Rs. } 40}$ and Hari gets $\boxed{\text{Rs. } 60}$.

• i) $\frac{(3m)^4 \times (3m)^{-3} \times (3m)^{-5}}{(3m)^{-2} \times (3m)^{-3}}$ Using the exponent rule $a^x \times a^y = a^{x+y}$: $$= \frac{(3m)^{4 + (-3) + (-5)}}{(3m)^{(-2) + (-3)}}$$ $$= \frac{(3m)^{4 - 3 - 5}}{(3m)^{-2 - 3}}$$ $$= \frac{(3m)^{-4}}{(3m)^{-5}}$$ Using the exponent rule $\frac{a^x}{a^y} = a^{x-y}$: $$= (3m)^{-4 - (-5)}$$ $$= (3m)^{-4 + 5}$$ $$= (3m)^1$$ $$= 3m$$ The simplified expression is $\boxed{3m}$.
• ii) $a^{a-b} \times a^{b-c} \times a^{c-a}$ Using the exponent rule $a^x \times a^y \times a^z = a^{x+y+z}$: $$= a^{(a-b) + (b-c) + (c-a)}$$ $$= a^{a-b+b-c+c-a}$$ $$= a^0$$ Using the exponent rule $a^0 = 1$: $$= 1$$ The simplified expression is $\boxed{1}$.

• $\frac{1}{4} + \left